Finite Group Databasebeta

← #21434910

Subgroups — #21434910

Order 7552.

466 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16837913C211yesmaximal, characteristic, complemented
2#16837732C211yesmaximal, characteristic, complemented
3#16837723C212yesmaximal, complemented
4#16837715C211yesmaximal, characteristic, complemented
5#16837653C212yesmaximal, complemented
6#16837646C212yesmaximal, complemented
7#16837553C211yesmaximal, characteristic, complemented
8#16837545C212yesmaximal, complemented
9#16837536C211yesmaximal, characteristic, complemented
10#16837366C211yesmaximal, characteristic, complemented
11C2×#121×C59C211yesmaximal, characteristic, complemented
12C2×#11745685C2211yescharacteristic, complemented
13C2×#11745670C2212yescomplemented
14C2×#11745669C2212yescomplemented
15C2×#11745664C2212yescomplemented
16#14273710—21no—
17#14273706C2211yescharacteristic
18#14273706C2211yescharacteristic, complemented
19#14273701C2211yescharacteristic
20#14273701C2211yescharacteristic, complemented
21#14273696—21no—
22C4×C22⋊D59—21no—
23C4×C22⋊D59—21no—
24#14273686C2211yescharacteristic, complemented
25#14273660C2212yescomplemented
26#14273658C2212yescomplemented
27#14273650C2212yescomplemented
28#14273648C2212yescomplemented
29#14273648C2212yescomplemented
30#14273647C2212yescomplemented
31C22⋊(C59⋊C8)C2211yescharacteristic
32C22⋊(C59⋊C8)C2211yescharacteristic
33C22⋊(C59⋊C8)C2212yescomplemented
34C22×D4×C59—21no—
35C22×D4×C59C2211yescharacteristic
36C23×C4×C59C2211yescharacteristic
37#121×C59—21no—
38#121×C59—21no—
39#121×C59C2212yes—
40#121×C59C2212yes—
41C2×C4⋊C4×C59C2211yescharacteristic
42C2×C22⋊C4×C59—21no—
43C24×C59—21no—
44C24×C59D411yescharacteristic
45C2×D4×C59—22no—
46C2×D4×C59—22no—
47C2×D4×C59—21no—
48C2×D4×C59—42no—
49C2×D4×C59—44no—
50C2×D4×C59C2312yes—
51C22×C4×C59—21no—
52C22×C4×C59—42no—
53C22×C4×C59D411yescharacteristic
54C22×C4×C59D411yescharacteristic
55C22×C4×C59D411yescharacteristic
56C22×C4×C59C2311yescharacteristic
57C22×C4×C59C2311yescharacteristic
58C22×C4×C59C2311yescharacteristic
59C2×C22⋊D59—21no—
60C2×C22⋊D59—21no—
61C2×C22⋊D59—21no—
62#11745685—42no—
63C2×D236C2311yescharacteristic
64C2×C4×D59—21no—
65C2×Dic118C2311yescharacteristic
66C4⋊C4×C59—21no—
67C4⋊C4×C59C2312yes—
68C22⋊C4×C59—22no—
69C22⋊C4×C59—22no—
70C22⋊Dic59—21no—
71C22⋊Dic59—21no—
72C22⋊Dic59—21no—
73#11745670—42no—
74#11745669—42no—
75(C2×D59)⋊C4—21no—
76(C2×D59)⋊C4—21no—
77C4⋊Dic59C2311yescharacteristic
78C4⋊Dic59C2311yescharacteristic
79Dic59⋊C4—21no—
80Dic59⋊C4—21no—
81C4×Dic59—21no—
82#11745664—42no—
83C2×C59⋊C8C2312yes—
84C2×C59⋊C8C2312yes—
85C23×C59—22no—
86C23×C59—22no—
87C23×C59—41no—
88C23×C59—41no—
89C23×C59—42no—
90C23×C59—42no—
91C23×C59—42no—
92C23×C59C2×D411yescharacteristic
93C23×C59C2×D411yescharacteristic
94C23×C59C2×D411yescharacteristic
95C22×D59—21no—
96D4×C59—22no—
97D4×C59—22no—
98D4×C59—42no—
99D4×C59—44no—
100D4×C59—44no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59—22no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—42no—
109C2×C4×C59C2×D411yescharacteristic
110C2×C4×C59C2×D411yescharacteristic
111C2×C4×C59C2×D411yescharacteristic
112C2×C4×C59C2411yescharacteristic
113C22⋊D59—42no—
114C22⋊D59—42no—
115C22⋊D59—42no—
116C2×Dic59—21no—
117C2×Dic59—21no—
118C2×Dic59—21no—
119D236—22no—
120C4×D59—42no—
121Dic118—22no—
122C59⋊C8—22no—
123C59⋊C8—22no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—21no—
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—44no—
133C22×C59—42no—
134C22×C59—42no—
135C22×C59—42no—
136C22×C59—84no—
137C22×C59AGL(1,ℤ/8)12yes—
138C22×C59C22×D411yescharacteristic
139C2×D59—41no—
140C2×D59—41no—
141C2×D59—41no—
142C4×C59—21no—
143C4×C59—21no—
144C4×C59—21no—
145C4×C59—42no—
146C4×C59H(5,𝔽2)12yes—
147Dic59—41no—
148Dic59—41no—
149Dic59—41no—
150#3194—591nomaximal, Sylow
151C2×C59—22no—
152C2×C59—41no—
153C2×C59—41no—
154C2×C59—82no—
155C2×C59#53411yescharacteristic
156C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
157C2×C59#57511yescharacteristic
158D59—81no—
159#525—591no—
160C2×#121D5911yescharacteristic, complemented
161#483—591no—
162#482—592no—
163#480—591no—
164#469—592no—
165#468—592no—
166#450—591no—
167#449—592no—
168#447—591no—
169#407—591no—
170C59#319411yescharacteristic, complemented, Sylow
171C2×#41—591no—
172C22×D4—21no—
173C22×D4C2×D5911yescharacteristic
174C23×C4C2×D5911yescharacteristic
175C2×SD16—592no—
176C2×D8—592no—
177C2×M16—592no—
178#123—1181no—
179C22⋊Q8—591no—
180C22⋊Q8—591no—
181#121—21no—
182#121—21no—
183#121—591no—
184#121—591no—
185#121C2×D5912yes—
186#121C2×D5912yes—
187#120—1181no—
188C4×D4—1181no—
189C4×D4—1181no—
190#117—591no—
191C2×C4⋊C4C2×D5911yescharacteristic
192C2×C22⋊C4—21no—
193#107—592no—
194#106—592no—
195#103—592no—
196#102—592no—
197#102—592no—
198#102—592no—
199C22⋊C8—591no—
200C22⋊C8—591no—
201C22⋊C8—592no—
202C24—21no—
203C24C22⋊D5911yescharacteristic
204#41—2362no—
205C2×Q8—591no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—591no—
212C2×D4—1181no—
213C2×D4—1181no—
214C2×D4—1181no—
215C2×D4C22×D5912yes—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—1181no—
219C22×C4C22⋊D5911yescharacteristic
220C22×C4C22⋊D5911yescharacteristic
221C22×C4C22⋊D5911yescharacteristic
222C22×C4C22×D5911yescharacteristic
223C22×C4C22×D5911yescharacteristic
224C22×C4C22×D5911yescharacteristic
225SD16—2362no—
226D8—2362no—
227M16—2362no—
228C2×C8—592no—
229C2×C8—592no—
230C4⋊C4—21no—
231C4⋊C4—591no—
232C4⋊C4—591no—
233C4⋊C4—1181no—
234C4⋊C4—1181no—
235C4⋊C4C22×D5912yes—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—1181no—
239C22⋊C4—1181no—
240C22⋊C4—1181no—
241C22⋊C4—1181no—
242C22⋊C4—1181no—
243C42—1181no—
244C23—22no—
245C23—22no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—1181no—
252C23C2×C22⋊D5911yescharacteristic
253C23C2×C22⋊D5911yescharacteristic
254C23C2×C22⋊D5911yescharacteristic
255Q8—1182no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—44no—
260D4—44no—
261D4—1182no—
262D4—2362no—
263D4—2362no—
264D4—2362no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—1181no—
274C2×C4—1181no—
275C2×C4—1181no—
276C2×C4—2362no—
277C2×C4C2×C22⋊D5911yescharacteristic
278C2×C4C2×C22⋊D5911yescharacteristic
279C2×C4C2×C22⋊D5911yescharacteristic
280C2×C4C23×D5911yescharacteristic
281C8—1182no—
282C8—1182no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—44no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—84no—
296C22—2361no—
297C22—2361no—
298C22—2361no—
299C22#1427375312yes—
300C22C22×C22⋊D5911yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—42no—
305C4—2361no—
306C4—2361no—
307C4—2361no—
308C4#1427377212yes—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—82no—
313C2—4721no—
314C2#1683796111yescharacteristic
315C2C2×#1427375311yescharacteristic
316C2#1683820011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×#121×C59C2
socleC22×C59C22×D4
hypercenterC2×#121D59
hypocenterC59#3194
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #16837913C2semidirect
2. #16837732C2semidirect
3. #16837723C2semidirect
4. #16837715C2semidirect
5. #16837653C2semidirect
6. #16837646C2semidirect
7. #16837553C2semidirect
8. #16837545C2semidirect
9. #16837536C2semidirect
10. #16837366C2semidirect
11. C2×#121×C59C2semidirect
12. C2×#11745685C22semidirect
13. C2×#11745670C22semidirect
14. C2×#11745669C22semidirect
15. C2×#11745664C22semidirect
18. #14273706C22semidirect
20. #14273701C22semidirect
24. #14273686C22semidirect
25. #14273660C22semidirect
26. #14273658C22semidirect
27. #14273650C22semidirect
28. #14273648C22semidirect
29. #14273648C22semidirect
30. #14273647C22semidirect
33. C22⋊(C59⋊C8)C22semidirect
160. C2×#121D59semidirect
170. C59#3194semidirect